notebook/notes/calculus/bounds.md

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Bounds Obsidian::STEM calculus::bounds
calculus

Overview

Suppose S is a nonempty set of real numbers and suppose there are numbers L and U such that L \leq x \leq U for all x \in S. Then S is said to be bounded below by L and bounded above by U. The number L is said to be a lower bound for S; the number U is said to be an upper bound for S.

%%ANKI Basic Let \varnothing \subset S \subseteq \mathbb{R}. What does it mean for S to be bounded below by B? Back: For all x \in S, B \leq x. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What does it mean for \varnothing \subseteq \mathbb{R} to be bounded above by B? Back: N/A. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Suppose S \subseteq \mathbb{R} is bounded below by B. What kind of mathematical object is S? Back: A nonempty set. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Suppose S \subseteq \mathbb{R} is unbounded above. What kind of mathematical object is S? Back: A set (possibly empty). Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Is the set of positive real numbers bounded below? Back: Yes. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let \varnothing \subset S \subseteq \mathbb{R}. What does it mean for S to be bounded above by B? Back: For all x \in S, x \leq B. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Is the set of positive real numbers bounded above? Back: No. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What are the upper bounds of interval [0, 1] \subseteq \mathbb{R}? Back: All real numbers x \geq 1. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What are the upper bounds of interval (0, 1) \subseteq \mathbb{R}? Back: All real numbers x \geq 1. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What are the lower bounds of interval [0, 1] \subseteq \mathbb{R}? Back: All real numbers x \leq 0. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What are the lower bounds of interval (0, 1) \subseteq \mathbb{R}? Back: All real numbers x \leq 0. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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If L \in S, then L is the minimum element of S (denoted L = \mathop{\text{min}} S). Likewise, if U \in S, then U is the maximum element of S (denoted U = \mathop{\text{max}}S). A set with no lower bound is said to be unbounded below. A set with no upper bound is said to be unbounded above.

%%ANKI Basic What is a maximum element of set \varnothing \subset S \subseteq \mathbb{R}? Back: A member of S that is also an upper bound. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is the maximum element of interval [0, 1] \subseteq \mathbb{R}? Back: 1. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is the maximum element of interval (0, 1) \subseteq \mathbb{R}? Back: N/A. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is a minimum element of set \varnothing \subset S \subseteq \mathbb{R}? Back: A member of S that is also a lower bound. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is the minimum element of interval (0, 1) \subseteq \mathbb{R}? Back: N/A. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is the minimum element of interval [0, 1] \subseteq \mathbb{R}? Back: 0. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is a maximum element of \varnothing \subset S \subseteq \mathbb{R} denoted? Back: As \mathop{\text{max}} S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is a minimum element of \varnothing \subset S \subseteq \mathbb{R} denoted? Back: As \mathop{\text{min}} S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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Least Upper Bounds

A number B is called a least upper bound (or supremum) of a nonempty set S if B is an upper bound for S and no number less than B is an upper bound for S. This is denoted as B = \mathop{\text{lub}}S or B = \mathop{\text{sup}} S.

%%ANKI Basic Let \varnothing \subset S \subseteq \mathbb{R}. What is a least upper bound of S? Back: An upper bound B for S such that no number less than B is also an upper bound for S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is an alternative term for a least upper bound of S? Back: A supremum of S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is a least upper bound for S denoted? Back: As \mathop{\text{lub}} S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is a supremum for S denoted? Back: As \mathop{\text{sup}} S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What distinguishes a supremum from a least upper bound? Back: They are synonyms of one another. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What distinguishes a supremum from a maximum? Back: A supremum is not necessarily a member of the reference set. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is the least upper bound of interval [0, 1] \subseteq \mathbb{R}? Back: 1. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

%%ANKI Basic What is the supremum of interval (0, 1) \subseteq \mathbb{R}? Back: 1. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How many upper bounds can a nonempty subset of \mathbb{R} have? Back: 0 or more. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How many supremums can a nonempty subset of \mathbb{R} have? Back: 0 or 1. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Is a maximum of a nonempty subset S of \mathbb{R} a supremum of S? Back: Yes. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Is a least upper bound of a nonempty subset S of \mathbb{R} a maximum of S? Back: Not necessarily. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S \subseteq \mathbb{R} have a supremum. If h > 0, why does there exist an x \in S such that x > \mathop{\text{sup}} S - h? Back: Otherwise \mathop{\text{sup}}S - h is an upper bound less than \mathop{\text{sup}}S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S \subseteq \mathbb{R} have a supremum. If h > 0, why does there exist an x \in S such that x < \mathop{\text{sup}} S - h? Back: N/A. This is not necessarily the case. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S \subseteq \mathbb{R} have an infimum. If h > 0, why does there exist an x \in S such that x > \mathop{\text{inf}} S + h? Back: N/A. This is not necessarily the case. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S \subseteq \mathbb{R} have an infimum. If h > 0, why does there exist an x \in S such that x < \mathop{\text{inf}} S + h? Back: Otherwise \mathop{\text{inf}}S + h is a lower bound greater than \mathop{\text{inf}}S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let A, B \subseteq \mathbb{R} have supremums. What set C satisfies \mathop{\text{sup}}C = \mathop{\text{sup}}A + \mathop{\text{sup}}B? Back: C = \{a + b \mid a \in A, b \in B\} Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let A, B \subseteq \mathbb{R}. When is \mathop{\text{sup}} \,\{a + b \mid a \in A, b \in B\} defined? Back: When A and B both have a supremum. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let A, B \subseteq \mathbb{R}. When is \mathop{\text{inf}} \,\{a + b \mid a \in A, b \in B\} defined? Back: When A and B both have an infimum. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let A, B \subseteq \mathbb{R} have infimums. What set C satisfies \mathop{\text{inf}}C = \mathop{\text{inf}}A + \mathop{\text{inf}}B? Back: C = \{a + b \mid a \in A, b \in B\} Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. Does S have a supremum? Back: Yes. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. Does T have a supremum? Back: Indeterminate. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. Does S have an infimum? Back: Indeterminate. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. Does T have an infimum? Back: Yes. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let S, T \subseteq \mathbb{R} be nonempty sets such that \forall s \in S, \forall t \in T, s \leq t. How does \mathop{\text{sup}} S compare to \mathop{\text{inf}} T? Back: \mathop{\text{sup}}S \leq \mathop{\text{inf}}T Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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Completeness Axiom

Every nonempty set S of real numbers which is bounded above has a supremum; that is, there is a real number B such that B = \mathop{\text{sup}} S.

%%ANKI Basic What does the completeness axiom of real numbers state? Back: Every nonempty set of real numbers bounded above has a supremum. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Consider \varnothing \subseteq \mathbb{R}. Why doesn't the completeness axiom of real numbers apply? Back: It only applies to nonempty sets. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Consider (0, 1) \subseteq \mathbb{R}. Why doesn't the completeness axiom of real numbers apply? Back: N/A. It does. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Consider \mathbb{R}^+. Why doesn't the completeness axiom apply? Back: It only applies to nonempty sets that are bounded above. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Consider (-\infty, 0). Why doesn't the completeness axiom apply? Back: N/A. It does. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What arbitrary decision was made when formulating the completeness axiom of real numbers? Back: Whether to use supremums or infimums. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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Greatest Lower Bounds

A number B is called a greatest lower bound (or infimum) of a nonempty set S if B is a lower bound for S and no number greater than B is a lower bound for S. This is denoted as B = \mathop{\text{glb}} S or B = \mathop{\text{inf}} S.

%%ANKI Basic Let \varnothing \subset S \subseteq \mathbb{R}. What is a greatest upper bound of S? Back: A lower bound B for S such that no number greater than B is also a lower bound for S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is an alternative term for a greatest lower bound of S? Back: An infimum of S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is a greatest lower bound for S denoted? Back: As \mathop{\text{glb}} S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is an infimum for S denoted? Back: As \mathop{\text{inf}} S. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What distinguishes a greatest lower bound from an infimum? Back: They are synonyms of one another. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What distinguishes an infimum from a minimum? Back: A supremum is not necessarily a member of the reference set. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What is the infimum of interval [0, 1] \subseteq \mathbb{R}? Back: 0. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

%%ANKI Basic What is the greatest lower bound of interval (0, 1) \subseteq \mathbb{R}? Back: 0. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How many lower bounds can a nonempty subset of \mathbb{R} have? Back: 0 or more. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

END%%

%%ANKI Basic How many greatest lower bounds can a nonempty subset of \mathbb{R} have? Back: 0 or 1. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Is an infimum of a nonempty subset S of \mathbb{R} a minimum of S? Back: Not necessarily. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Is a minimum of a nonempty subset S of \mathbb{R} a greatest lower bound of S? Back: Yes. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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Bibliography

  • Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).